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Ratio of circumscribed triangle to inscribed triangle

Equilateral triangle ABC has a circle inscribed in it. Inside the circle is inscribed another equilateral triangle, DEF. What is the ratio of AB to DE? (Answer from back of book: 4 to 1). I would expect the answer to be 2, but we don't know how to get the answer.

I tried make a picture of what I mean (attached), but the letters came out wrong.

**Edit**: Ignore the point labels in the figure. I need the ratio of one side of the big triangle to one side of the little triangle. Thanks.

Re: Ratio of circumscribed triangle to inscribed triangle

Mathdad

your figure is not correct. you put D at the barycenter of the triangle....

Please give us more accurate information in order to help you.

if you are searching for a ratio between the initial triangle ABC and the one inscribed then the ratio of their areas is 4:1

Re: Ratio of circumscribed triangle to inscribed triangle

I know, but I am not skilled enough with Geogebra to change the letters. I said that in my first post, but thought I included enough information to make it clear what I wanted.

I need the ratio of one side of the big triangle to one side of the small triangle. Thanks.

Re: Ratio of circumscribed triangle to inscribed triangle

Quote:

Originally Posted by

**mathDad** I know, but I am not skilled enough with Geogebra to change the letters. I said that in my first post, but thought I included enough information to make it clear what I wanted.

I need the ratio of one side of the big triangle to one side of the small triangle. Thanks.

$\displaystyle m(\overline{EH})=\frac{m(\overline{AB})}{2}$.

$\displaystyle m(\overline{ED})=\frac{m(\overline{AE})}{3}$

$\displaystyle m(\overline{AE})=\frac{\sqrt{3}m(\overline{AB})}{2 }$

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Re: Ratio of circumscribed triangle to inscribed triangle

Hi Dad,

I hope the attached diagram answers your question:

Attachment 27619

Re: Ratio of circumscribed triangle to inscribed triangle

Quote:

Originally Posted by

**johng**

There is a problem with you diagram.

It is not labeled the same as the one on the OP.

Your O is his D.

He did ask about $\displaystyle \overline{ED}$.

Now, I will grant you that there seems to be a contradiction in his post.

Re: Ratio of circumscribed triangle to inscribed triangle

Quote:

Originally Posted by

**johng** Hi Dad,

I hope the attached diagram answers your question:

Yes, very nice. Just the way I thought (the book must have a typo).

Re: Ratio of circumscribed triangle to inscribed triangle

Quote:

Originally Posted by

**Plato** There is a problem with you diagram.

It is not labeled the same as the one on the OP.

Your O is his D.

He did ask about $\displaystyle \overline{ED}$.

Now, I will grant you that there seems to be a contradiction in his post.

Because the diagram in my OP is not correct. My bad.